Low Earth Orbit (LEO) satellite communication networks have become an important means to provide internet access services for areas with limited infrastructure. Compared with the Geostationary Earth Orbit (GEO) satellites, the LEO satellites have limited on-board communication caching and calculating resources. Furthermore, the distribution of traffic requests is dynamically changing and uneven due to the relative movement between the LEO satellites and the ground. Therefore, how to schedule the multi-dimensional resources is an important issue for the LEO satellite communication networks. Beam-hopping is an efficient approach to improve the resource utilization by dynamically allocating time, power, and frequency according to the traffic requests. This paper proposes an efficient multi-dimensional resource allocation mechanism for beam-hopping in LEO satellite networks, which simultaneously satisfies the GEO interference avoidance. First, we construct the beam-hopping model of LEO satellites, and formulate the resource optimization problem. Second, we provide the weighted greedy strategy to determine the illumination pattern. In order to reduce the search space, the cells are clustered to non-interference clusters. Then, an improved genetic algorithm is provided to jointly allocate the communication resources. Finally, we construct various simulations to evaluate our proposed mechanism. Compared with the random-BH, polling-BH and traditional genetic algorithm, our algorithm achieves better performance in terms of both system throughput, access success rate, average delay and fairness between cells. The performance improvement is more significant in scenarios where traffic demand is unevenly distributed.